Local isometric imbeddings of symplectic groups (Q1392732)

Let \(G\) be a simple compact Lie group, \(\mathfrak g\) its Lie algebra, \(\mathfrak h\) a Cartan subalgebra of \(\mathfrak g\). Denote by \(p_G\) the maximum of the dimensions of subspaces \(W\subset\mathfrak g\) satisfying \([W,W]\subset\mathfrak h\). The authors prove that \(p_{\text{Sp}(n)} = 2n\), \(n\geq 1\). Using the results of their previous paper [\textit{Y. Agaoka} and \textit{E. Kaneda}, Hiroshima Math. J. 24, 77-110 (1994; Zbl 0808.53050)], they deduce that the canonical imbedding of the group \(\text{Sp}(n)\), endowed with the bi-invariant Riemannian metric, into the vector space \(\mathbb{R}^{4n^2}\) of the quaternion \((n,n)\)-matrices is the least-dimensional isometric imbedding (even from the local standpoint). They also prove that \([3n/2]-1\leq p_G\leq 2n-1\) for \(G = \text{SU}(n)\), \(2n\leq p_G\leq 4n+1\) for \(G = \text{SO}(2n+1)\), \(n + 2[n/2]\leq p_G\leq 4n-1\) for \(G = \text{SO}(2n)\), and they give the corresponding estimates on the dimensions of local isometric imbeddings of these compact Lie groups.